Let denote the set of all permutations of . Let and denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in . Let denote the set of subsequences of with length at least three. Let denote .

A permutation is called a Roller Coaster permutation if . Let be the set of all Roller Coaster permutations in .

Conjecture For ,

\item If , then . \item If , then with .

Conjecture (Odd Sum conjecture) Given ,

\item If , then is odd for . \item If , then for all .

Bibliography

*[AS] Tanbir Ahmed, Hunter Snevily, Some properties of Roller Coaster permutations. To appear in Bull. Institute of Combinatorics and its Applications, 2013.